PHYSICAL REVIEW A VOLUME 14, NUMBER 5 NOVEMBER 1976

Constant-coupling theory of nematic liquid crystals

Ping Sheng and Peter J. WojtowiczRCA Laboratories, Princeton, New Jersey 08540

(Received 4 June 1976)

The mell-known constant-coupling theoryof ferromagnetism has been adapted to the problem of the nematicliquid crystal. A cluster variational formulation of the theory is developed. The calculations are based onaneffective pair potential which is derived from the most general form of interaction by taking suitable averages.Expressions for the long-range order, short-range order, energy, and free energy are derived. Numericalcalculations have been performed for a variety of effective interaction potentials. The temperature dependencesof the order parameters and thermodynamic functions have been determined. As required by experiment, thetheory is found to display a first-order nematic-isotropic phase transition. The results of the present theory arecompared to those of previous calculations; questions concerning the validity of the mean-field theory havebeen clarified. The numerical results are also compared to a variety of experimental data.

I. INTRODUCTION

The statistical mechanics of nematic j.iquid cry-stals is not nearly as advanced as the correspond-ing theories of other ordered phases. The statisti-cal mechanics of the va, rious magnetic phases andof their critical behavior, for example, is veryhighly developed. For nematic liquids on the otherhand, the only theories available include the hard-rod model, "various forms of the mean-fieM ap-proximation, ' ' and some Monte Carlo calcula-tions. " In this paper we describe a new statisti-cal treatment of the nematic phase based on thewell known "constant- coupling" theory of ferro-magnets.

The constant-coupling theory was initially intro-duced by Kasteleijn and Van Kranendonk" to pro-vide an improvement over the Weiss mean-fieldmodels. The precise relationship of this theoryto the rigorous cluster expansion treatment offerromagnets was subsequently demonstrated byStrieb, Callen, and Horwitz. " Numerous ap-plications of the theory to problems in ferromag-netism and antiferromagnetism have appeared, ""and in all eases the results are found to be morerepresentative of real magnets than those givenby the mean- field approximation. The eonstant-coupling trea, tment provides more a,ccurate esti-mates of the critical points, gives a more realis-tic temperature dependence of the magnetization,heat capacity, and high- temperature susceptibility,and is capable of accounting for many of the ef-fects of short- range order.

With its basis in statistical mechanics thusfirmly established, the constant-coupling theoryappears to be a natural candidate for the extensionof the theory of nematic liquid crystals beyond themean-field approximation. In Sec. II we will de-velop a cluster va, riational approach to the con-stant-coupling theory of the nematie phase by fol-

lowing the formulation of this method given bySma, rt." The effective pair interaction potentialused is derived from a general form of inter-molecular pair potential by taking suitable aver-ages. Expressions for the long-range order,short-range order, energy, and free energy arethen derived and the self-consistency of the theoryis explored.

In Sec. III we describe the results of the theoryobtained by numerical solution of the equations fora variety of effective interaction potentials. Thetemperature dependence of the order parametersand thermodynamic quantities are of special in-terest. As required by experiment, the theory isfound to display a first-order nematic to isotropieliquid-phase transition.

Section IV is devoted to a comparison of theresults of this theory to those of the mean-fieldtreatxnent, the Monte Carlo calculations, andseveral other models. The difficulties encounteredin other attempts'"" at the cluster variationalapproach are also examined and resolved. We con-clude this paper in Sec. V by comparing ournumerical results to experiments on some realnematic materials.

II. FORMULATION

A. Pair-interaction potential

The most general form of pair-interaction po-tential v» between two axially symmetric mole-eules can be written in. the form of the Pople ex-pansion"

v„(r, (9„@„8„@,)

=4m Q Q Qu~ ~ (r)Il-o I2-0 m

"&&,.(~l, vD~*...(~l, wl) .

(&)

1884 PING SHENG AND PETER J. WOJTOWICZ 14

Here r is the distance of separation between thecenters of the two molecules; 0,', 0,', y,', y2' are thepolar angles of the two molecules defined as shown

in Fig. 1; Y~ (8, rp) is the usual spherical har-monic; * denotes complex conjugation; and u~ ~, (r)are functions of r that vanish for ]m ~)L„L,Since v» is a real and even function of y,' —y2', thefunctions u~ ~ (r) have the property that uz ~ (r)=u.~ ~ (r).

A rigorous theory of a fluid system based on thegeneral form of pairwise interaction potential ofEq. (1) is impossibly difficult. A simpler approachis to derive an "effective" pairwise interactionpotential which can be obtained from Eq. (1) byperforming certain averages. That "effective"pairwise interaction potential will then be thestarting point of the theory presented in this paper.

In order to derive an effective interaction po-tential, we first express v» in terms of a polarcoordinate system based on the director of thenematic phase n as the polar axis. The coordinateaxes for the molecules 1 and 2 must be rotatedfrom that shown in Fig. 1(a) to that shown in Fig.1(b). The unprimed angles now describe theorientations of the molecules with respect to thenew rotated coordinate system. In the new co-ordinate system v» has the form

v12 = 4& +L&L2, & DP10 L20 m

x Y,(8„y,)(D,„)*

x Yz,(8„y,),where the D~~ are the elements of the Wigner ro-tation matrices. " Now we take two successiveaverages of v„. First, we average over all orien-tations of the intermolecular vector r. Next, weaverage v» over all values of the intermolecularseparation r. The combination of these two aver-ages will remove all dependences of v12. On theintermolecular vector r. Such a double average

(denoted by (( ) ) ) has been carried out by Hum-

phries et al. ' The resulting effective interactionpotential V» can be expressed as a series in

Legendre polynomials PL:

V„=—((v„))= g CIPI(cos8») .

v0 n L PL cos0»L(even)

(4)

It should be noted that n2 -=1 and v, & 0 by definition.

B. Cluster variational approach

Our task now is the formulation of a theory sothat the thermodynamic properties of the systemcan be calculated from V». The simplest ap-proach is the mean-field theory, which would

further reduce V» into a one-particle potential(denoted as V„) by averaging over a.ll possibleorientations of molecule 2. Since the single-particle orientational distribution is expected tobe axially symmetric about the director n and

therefore independent of y, such an average yields

V =- V„=—v, Q o.~ (P~) P~(cos8, ),L(even)

where 0, is the angle of molecule 1 with respectto the director n; and (P~), the average value of

PL, can be regarded as the order parameters ofthe problem. If each molecule has z neighbors,then the total mean field felt by the molecule isz V„r. Given the form of V„F as in Eq. (5), theorder parameters (P~) can be determined by theself- consistent equations

Here CL's are constants, and 0» is the relativeangle between the long axes of molecules 1 and 2.In the following we will assume that C, &0 and thatthe interaction potential V» cannot distinguish be-tween the heads and tails of the molecules, which

implies that only the even L terms will be includedin the sum. If we further define v, = ~C, ~, n~—= C~/C„ then V» can be put in the form

. /I,

8 f, P~(cos8, ) exp[ Pz VMr(cos8, ) ]d(cos8, )

f ' exp[ pz V„r(c—os8, )] d(cos8, )

(b)

FIG. 1. Coordinate systems required to describe theinteraction between two axially symmetric molecules:(a) the intermolecular vector x is the mutual polar axisand (b) the director n is the polar axis for each mole-cule.

where P —= I/kT, k being Boltzmann's constant and

T the temperature.In another development, Qguchi" has invented

a method in which a pair of molecules interactingwith V» is placed in an effective mean field ofits neighbors. If each molecule has z neighbors,then the total interaction energy of the pair isgiven by

14 CONSTANT-COUPLING THEORY OF NEMATIC LIQUID CRYSTALS

V~~——(a —1)[VMF(cosg,)+ VMr(cos82)]+ V,2

= —V eL Z —1 PLL {even)

x [P~(cos8,) + P~(cos 8,) ]

+ Pg(cos8~2) j,

where cos8» = cos8„cos8,+ sing, sing, cos(y, —p,}.In the Qguchi method the long- range-order param-eters (P~) are determined by the consistency con-ditions

fo f, f, P~(x) exp(- P Vo,„,h;) d$ dx dy

f'f' f' exp(-PV~„„;)dgdxdy

where x:—cosg„y = cos82, and $ —= (y, —rp, )/2wThe Qguchi method also yields short-range-orderparameters 7L:

fo fo fo P~(cosg„) exp(- PVo ~;)d(dxdyf'f' f' exp(- P Vo,~) d) dxdy

(9)

The cluster variational method can be obtainedby synthesizing the mean-field theoxy and theQguchi method. %e assume that the general formof the mean field as felt by molecule 1, say, canbe written

V„r(cosg, ) =- v, P o~[T, (P~)]P~(cosg,), (10)L {even)

where &r~[T, (Pg] is a general function of T and

(P~)(V„v specializes to V„r if a~= n~(P~)) In.order to determine the value of oL at every tem-perature T, we note that given V„F, there aretwo ways to calculate (P~). One way is to use themean-field theory, Eq. (6), where in place of V„rwe substitute V„~. Another way is to use theQguchi method in which the total interaction en-ergy is now given by

Vo~,~ = (a —1)[VMr(cosg, ) + VMr(cosg, ))+ V„

g (z —1)o~ [P~ (cos8,)+ P~(cosg, )]+o'~ P~(cosg») . (11)L(even)

(P~) can then be determined by using Eq. (S) with Vo~,h; replaced by Vo~,„;. Following Smart, "the con-dition that we are going to impose on the o's is that the two ways of determining (P~) have to yield identicalresults. That is,

1 1p(*)exp'Esp()drjsxp'zgap(x))d*0 L(even) 0 L(even)

1 1 1 pPL x exp 0 ~ —~ oL PL x +PL & + ~LPL cos~12

0 0 0 L(even) L(even)

1 1 1 I 10 z 1 OL PL x +PL y + nLPL cos~12 d$dxdy

0 0 0 L(even) L(even)

(12)

Here cosg„=xy+ [(1—x')(1 —y')]'~' cos2vg. Equa-tion (12) is the heart of the cluster variational ap-proach. There is one equation for every even L.In every one of these equations ~ and nL's are theinputs and oL's are the unknowns. Simultaneoussolution of all the equations yields the dependenceof o~'s on the reduced temperature kT/v, . Oncethe crL are known, the long- range-order param-eters, (Pg and the short-range-order param-eters, ~L, can easily be determined in the samemanner as in the Qguchi method. Namely, we canuse Eqs. (8) and (9}with Vo,~ replaced by Vo,„d„.

C. Statisticai thermodynamics

In order to determine the internal energy of thesystem in which the pairwise interaction between

the molecules is given by V», we note that the in-ternal energy for a. single pair E~ is given by

Ep= —'U0 QL PL cos6}12 = —50 QL TL .L(even)

Here ( ) denotes thermodynamic averaging. Ifeach molecule has z neighbors, then for eachmolecule we count z pairs of interactions. There-fore, the total internal energy E of the system isgiven by

1 1E= g NZEP ———P V0NZ nL 7L,L(even)

where N is the total number of molecules and thefactor & arises because each interaction has been

PING SHENG AND PETER J. %03TOWICZ

counted twice.Next, we write down the free energy F of the

system. The free-energy function must satisfytwo requirements: (i) Setting BF/Bo~=0 shouldgive Eq. (12), the consistency condition for the o~.(ii) B(pP)/Bp, evaluated at those values of oz whichsatisfy Eq. (12), should yield E Th. e first require-

ment follows from the fact that the solutions de-termined from Eq. (12) must be those that repre-sent the extrema of free energy. The second re-quirement follows directly from thermodynamics.It is easily vex'ified that a free-energy functionwhich satisfies these two requirements can bewx itten

NAT=(z —1) ln

1

exp ' z Q a~P~(x) dxuT L (even}

)

exp (* —1) E )p ( ) p (7))+ E p ( osp ) )) x)dVI+c 08t.0 'kT I (even} 1.(even}

The question arises as to whether there areother free-energy functions which satisfy thesame two requirements. In other words, are thetwo requirements stated above sufficient to unique-

ly determine the free-energy function& In theappendix we show that the two conditions are onlysufficient to determine (to within a constant) thefree energy PF at those values of cr~'s whichsatisfy Eq. (12), denoted as oz(T). Therefore,the answer to the uniqueness question is that upto a constant, the values of PF as given by Eq.(15) is unique over the domain of crQ~(T).

III. NUMERICAL CALCULATIONS

In this section we present results of numericalcalculations based on an interaction potential inwhich only the leading term of V» is retained.Namely, we will let

equations will have n, = 1 and all the other n~'sequal to zero. The values of v~'s at each reducedtemperature kT/vQ can then be obtained by thesimultaneous solution of all the equations. How-

ever, from the mean-field theory it is expectedthat if V» is of the form expressed by Eq. (16),then the mean field experienced by molecule 1would mostly be of the form P, (cos8,). That is, o,is expected to be much larger than all the other0~'s. This expectation is borne out by calculationsas shown in a later section. Therefore, for sim-plicity we will retain only &r, and o, in Eq. (12) andset all the other 0~'s equal to zero. In this man-ner we have reduced the problem to two simul-taneous equations for the determination of cr, and0'4.

P( )xf, ( )xdx

12 Q 2( 12) '

The effect of including one higher-order term,P,(cos8»), in V„will also be investigated.

(16)

P~(x) f, (x) dx

P, (x) f,(x, y, () dx dy d$, (17a)

A. Thermodynamic properties

Theoretics. lly, V„given by Eq. (16) can be di-rectly substituted into Eq. (12). The resulting where

1 1 1

P, x f, x, y, ( dxdyd$, 17b0 0 0

)exp((v, /k 7')z[a,P, (x) + o,P,(x) ]jj' expovQ/k7')z [o,P, (x) + o, P,(x)]]dx '

exp/(vQ/k T) [(z—1)[o,P, (x) + o,P, (y ) + o~P, (x) + v4PQ(y)]+ P, (cos8»)]jj' j' j' exp/(vQ/kT)[(z —1) [o,P, (x)+ &&,P, (v) + p,o( )+xo,P,(y)]+ p, (cos8„)]]dx

dydee

The solution of Eqs. (17a) and (17b) for a given zyields values of a, and cr, at each reduced temper-ature kT/vQ. Once these values are known, wecan calculate the long- range-order parameters

(P,) and (P,):1

(P, ,) = P, ,(x)f, (x) dx.0

(19)

14 CONSTANT-COUP LING THEORY OF NEMATIC LIQUID CRYSTALS 1887

The short-range order ~, is obtained similarly,

T2= f1 1 1

P, (cose») f,(x, y, f) dx dy dg.0 0

(20)

The free energy can be obtained by using Eq. (15),which can be cast in the form

F 1—=-—,'

v ozr, —kT (z —1) f, (x) lnf, (x) dxN 0

TABLE I. Constant-coupling theory results using thepotential of Eq. (16). (P2)c is the long-range-orderparameter at the transition temperature &c, while 72"(&

andy2 are the short-range-order parameters of thenematic and isotropic phases, respectively, at Tc. DSis the entropy change at the transition. The column forz = ~ represents the results of the mean-field theory(Ref. 4j.

10

+ const.

~"f.(*,w, 8 d*&v &t)

(21)

(P2~ c

(n)2

~( i)2

0.356

0.393

0.320

0.382

0.298

0.189

0.399

0.264

0.134

0.405

0.245

0.103

0.429

ZS/N 0.210 0.282 0.323 0.346 0.418

kI'c /zv 0 0.1740 0.1933 0.2011 0.2055 0.2202

where the first term is the internal energy perparticle and the second term is the entropy con-tribution to the free energy.

It should be noted that v„cr, = 0 is a solution ofEqs. (17a) and (17b) for any value of kT/v, . Thisis the disordered phase, the isotropic liquid. Atlarge values of kT/v, this is the only solution.However, below a certain reduced temperature twomore solutions appear. The equilibrium solutionin that temperature range is then determined bythe criterion of minimum free energy. The cal-culation is done on a computer using Gaussian in-tegration techniques and Newton's method in lo-cating the solutions. In Fig. 2 we present thevariation of (P,) and r, vs kT/v, for z= 6. Theaccurate values of (P,) and 7, at the transition,the transition temperature, and the entropy dis-continuity ~S are given in Table I for z = 10, 8, 6, 4.The results for z =~, the mean-field theory, arealso presented for comparison. Since in liquidcrystals z is the "average number" of nearest

neighbors, z can take on nonintegral values. InFigs. 2 and 4 we show the variation of k T, /v, andthe transition entropy dS/Nk as a function of z,treated here a,s a continuous variable. It shouldbe noted that a.s a function of z, the nematic-isotropic phase transition tends to be more"second-order-like" as z decreases. Morespecifically, the discontinuities in (P,) and r, atthe transition become smaller, and ~S decreasesrapidly as z decreases. The vanishing of transi-

2.4—

2.2—

20—

18—

Z$6

14—kTc

12—

10-

.4—

. 2

o I I I I I I I I

4 .5 6 .7 .8 .9 I.O I. I 1.2 1.5 1.4 I,5 1.6kT /vo

0I 2 3 4 5 6 7 8 9 10

FIG. 2. Temperature variation of long- and short-range order for z =6. (P2) is the solid curve while T2

is the dashed curve.

FIG. 3. Dependence of the first-order phase transi-tion temperature Tc on effective nearest-neighbor num-ber z.

PING SHENG AND PETER J. %OJTOWICZ 14

.8-

0.40—MEAN FIELDTHEORY

=0.2

0.30-

020—

kTc/vo = 1.687

I I I I I I I I

.8 .9 1.0 I. I 1.2 I 5 1.4 1.5 1.6 1.7 l.e I.9 2.0kTc/ve

o. lo—

FIG. 6. Temperature variation of the long- and short-range order for z =8 with n4 —+0.2. (P2) is the solidcurve while T'2 is the dashed curve. The dot-dashedcurve is g'2) for z = 8 and +4=0 for comparison.

B. Effect of the P4 term in intermolecular interaction

tion temperature at z =2 is understandable sincethe system has to be one-dimensional for z =2,and it is a vrell-known fact that one-dimensionalsystems cannot have phase transitions at finitetemperatures. In Fig. 5 we show the variation ofthe specific heat C,

C z deNk 2 d(kT/vo)

'

as a function of temperature for z =6.

(22)

2 3 4 5 6 7 8 9 loz

FIG. 4. Dependence of the entropy discontinuity 68on effective nearest-neighbor number z.

The calculation mentioned above can easily beadapted for a V» of the form

V» = —II,[P,(cos8»)+ a, P,(cos8»)] . (23)

We replace P,(cos8„) in Eq. (18b) by P,(cos8»)+ &, P(c os8») Then t.he solution of Eqs. (17a) and

(17b) yields values of o, and o, at every kT/v0 In.Figs. 8 and 7 we show (P,) and 7, as function ofkT/v, for z =8 and n, = +0.2. It is seen that thevalues of transition temperature, (P,), and r, areall raised for a, =0.2 and are depressed for e,= —0.2. (P,) for o.,=0 is also plotted for com-parison.

Z=6

.8-Z= 8

o4= -0.2

75-

I

0 I I I I I I I I I I

.5 6 .7 .8 .9 I.Q I. I 1.2 I.g 1,4 1.5 1.6kT/ve

FIG. 5. Temperature variation of the heat capacityfor z =6.

k Tc/ve = I. 54 5

0 I I I I I I I I

.8 .9 I.Q I.I l.2 1.3 l.4 I.5 1.6 1,7 I.S I.9 2.0kT /ve

FIG. 7. Temperature variation of the long- and short-range order for z =8 with n4-——0.2. {P2) is the solidcurve while 72 is the dashed curve. The dot-dashedcurve is (P&) for z = 8 and n4 —-0 for comparison.

14 CONSTANT-COUPLING THEORY OF NEMATIC LIQUID CRYSTALS 1889

C. Paranematic susceptibility

When a magnetic field is applied to a nematicliquid crystal in its isotropic phase, a smallamount of nematic order will be induced due tothe diamagnetic anisotropy of liquid crystal mole-cules. The amount of order is proportional to thesquare of the magnetic field

magnetic anisotropy for a single molecule. Thispotential has to be added to the mean field eachmolecule experiences from its neighbors. There-fore, in the self-consistency condition, Eq. (12),we have to replace

' Z (JLPL XkT

L(even)

(P,) =qH', (24) by

)1~(T- T,*) ', (25)

where T,* is a temperature lower then T„ the tran-sition temperature, and can be interpreted as thetemperature below which the supercooling of theisotropic phase becomes impossible. In the mean-field theory we have the relation

T,*=0.9082 T, . (26)

Experimentally, T,* is about 0.99 T, . Therefore,there is a wide discrepancy between the mean-field theory and the experimental result. In thissection we examine the temperature dependenceof g and calculate the magnitude of T,* in the con-stant- coupling theory.

When a magnetic field H is applied, each mole-cule experiences an orienting potential of the form

W„=——,H'Ay P, (cose), (27)

where 8 is the angle between the long axis of themolecule and the field direction, and ~y is the dia-

where the proportionality constant g always in-creases in magnitude as the transition temperatureis approached from above. In fact, in the mean-field theory and the Landau-deGennes theory g isgiven as

L(even) 0

on the left-hand side of the equation and replace

(z 1) Q a~[P~(x)+Pz(y)]L(even)

(z —1) g og(Pg(&)+Pg(y)]L(even)

H'&y(tP, (x)+ P, (y)]3vp

(, ). (... *y)

IV0 H 4y2kT ' 3Vp

with

(28)

on the right-hand side of the equation. Since thequantity H'd. y/v, is usually small for H& 10' Oe,we can assume that the o~ will also be small (sincein the limit of H-0 all oL-0 in the isotropicphase). As a result exponentials of these quanti-ties can be expanded, yielding

1 1 1Vp

0 0J (y'. (*)~ y'. (y))* o oy E,y', (oo o,.))d(d dyL(even)

1 1 1vpX O' E P ( ooo„))d(d*dy

0 0 0 kT L (even)

(28)

From Eq. (28) it is easily calculated that

H'&X

3v, 5I —(5I —2)z' (30)

which q diverges, from Eq. (32) we obtain theequation

I(T,*)= '-, z/(z —1) .

Substitution of Eq. (30) into Eq. (28) yields

5I15k T 5I —(5I —2)z

Comparison of Eq. (31) with Eq. (24) gives

(31)

15kT „5I (32)

Since T,* can be defined as that temperature at

The values of I at various temperatures have beencalculated on computer for the case of V1p= —v,P, ( sgc»o). In Fig. 8 we plot 1/)7 a,s a functionof kT/v, for z =6. The curves are extended below

kT, /vo for the sake of illustration. It should benoted that 1/q is no longer a linear function of T,although the deviation from linearity is small.Also, we note that the value of T~ obtained fromEq. (33) is a function of z. For z =6 we have T,*

1890 PING SHENG AND PETER J. WOJTOWICZ 14

0.4— 1.02—

1.00

Z=B

0.01

0.3—

hX I

I5v, g

0.2—

.98

.96("&

.94

.92

—-0.02

—-0.034

CTp

—-0.04

—-0.05

0. 1

0

kTcYo

1.2 1.3kT/va

1.4 1.5

~/yo

~ 90' I I I I I I -0.06.9 1.0 I. I 1.2 1.3 1.4 1.5 1.6 1.9 2.0

kT/yo

FIG. 9. Temperature variation of the variationalparameters 02 and 04 for z = 8.

FIG. 8. Temperature variation of the reciprocalparanematic susceptibility 1/g for z = 6.

=1.097 v, /k=0. 9457T, and for z = 8 we have T,*

=1.510 v, /k=0. 9385T,. Therefore, in each casethe ratio of T, /T, has been ra. ised above that ofthe mean-field theory. However, the increaseis not enough to explain the experimental result.Similar improvements in T,*/T, have been obtainedby Madhusudana and Chandrasekhar" using a some-what different technique which also includes short-range- order effects.

D. Magnitude and the form of the mean field

It is seen in Sec. II that if V» ——v, P, (cos8»),then, in the mean-field theory, the mean field asexperienced by molecule 1 due to molecule 2 isgiven by

VMr = —vo(P, ) P, (cos8, ) . (34)

Two features of VM~ should be noted. First, themagnitude of V„r is proportional to (P,). Second,the form of VMF as a function of cos6), is given byP, (cos8,). In this section we examine the accuracyof these two approximations by using the constant-coupling theory.

In Sec. IIIA it is shown that in the constant-coupling theory the mean field as experienced bymolecule 1 can be approximated by

V„r = —v, [o,P, (cos8,)+o,P,(cos8,)], (35)

where the values of o, and 04 at each reduced tem-perature kT/v, are determined by Eqs. (17a),(17b), (18a), and (18b). It should be rememberedthat we have dropped all terms higher than

P, (cos8) in the expectation that they are small.In Fig. 9 we show the results of the calculationfor z =8. It is clear from the graph that v, is notproportional to (P,). The ratio of o,/(P, ) is adecreasing function of temperature, reaching thevalue of -0.92 at the transition. We also see that

the form of the mean field is not strictly P, (cos8, )but contains a. relatively small portion of P, (cos8, ).04 is always negative, and the absolute magnitudeof the ratio o, to o„~o,/o, ~, is a. decreasing func-tion of temperature, reaching -0.025 at the transi-tion. It should be noted that the sharp increaseof o,/(P, ) and ~o,/o,

~

below kT/v, = 1.1 may becaused by the omission of P,(cos8, ) and P, (cos8,)

terms in the series for V». However, the figureis indicative of the general trend for these quan-tities and offers an estimate of the accuracy ofthe mean-field approximation given by Eq. (34).

Before leaving this section, we point out that inall the numerical calculations the n~ and z havebeen treated as temperature-independent quan-tities. In the realistic case this assumption maynot be valid since in liquid crystals the constituentmolecules do not have fixed positions as in asolid, and as a result, the average environment(such as the average number of nearest neigh-bors) of a molecule may vary with the temperature.In view of such possibilities, the temperature in-dependence of the n~ and z should be regarded asa first-order approximation. Further study is re-quired to establish the relationships, if any, be-tween the a~ and z on the one hand and T, theshort- range order (and/or long- range order) onthe other.

IV. COMPARISON WITH OTHER THEORIES

Qualitative and quantitative comparisons of theresults of the constant coupling approach withthose of the mean-field theory have been giventhroughout the previous sections. With bothmethods the nematic-isotropic transformation isfound to be a first-order phase transition. Theconstant coupling theory, however, yields lowervalues for the transition temperatures, latentheats, entropy changes, and long-range-orderparameter discontinuities in all cases. In addition,the present calculation also provides a larger valuefor the parameter T,*/T,

An improvement over the mean-field theory was

14 CONSTANT-COUPLING THEORY OF NEMATIC LIQUID CRYSTALS 1891

attempted by Lebwohl and Lasher' using MonteCarlo techniques for the case of z = 6. A first-order phase change was observed here also. Thecritical temperature was found to be 1.124vo/k(in our notation), while the entropy change andlong-range-order discontinuity were 0.097Nkand 0.33, respectively. The critical temperaturecompares favorably with our value of 1.160',/k(Table I). The long- range- order discontinuity com-pares well with our value of 0.382 and is likewiselower than the mean-field result of 0.429. The en-tropy change obtained in the Monte Carlo calcula-tion, however, is much smaller than our value0.282Nk. In Sec. V we will see that the experi-mentally determined entropy changes comparefavorably with the constant coupling values.

Raich, Etters, and Flax" and Schultz" haveraised the question of whether the first-orderphase transition predicted by the mean-field mo-del is merely a spurious result of the mean-fieldapproximation. Based on a cluster variational ap-proach simila, r to ours but using a separable po-tential function [V»-—V(8,) V(8,), where the 8,. givethe orientations of the molecules with respect tosome external axisj, Raich et a/. "failed to obtaina phase transition of any kind. With a differentkind of statistical model, but again using a separ-able potential function, Schultz" also failed toobtain a, phase transition. Priest, "however, hasshown that the inability to observe a phase changeis caused by the adoption of a, sepa, rable form ofpotential. The separable form is unphysical inthat it destroys the rotational invariance of theoriginal potential function, and leads to qualita-tively different results. Taking a quantized modelof a nematic system but using a proper rotationallyinvariant pair potential, Priest" was able to re-cover the first-order phase transition.

In this paper we began by assuming a perfectlygeneral pair interaction potential and proceededto simplify it by making a set of reasonable as-sumptions which naturally retained the rotationalinvariance property. Using this simplified po-tential a,s the basis for the constant coupling cal-culations we then obtained first-order phasetransitions as required. The results of the mean-field theory, furthermore, can be approached fromthe present formulation by passing to the limit ofvery large z. We therefore conclude that the first-order transition of the mean-field theory is notspurious; the theory gives correct qualitative pre-dictions because it does indeed contain the basicelements of the physics of the problem. The situa-tion here is thus no different from the ca,se ofmagnetism where the relationship of the mean-field model to more general theory is well estab-lished.

V. COMPARISON WITH EXPERIMENT

7—I

Q I

.88 .89 90 .91 .92 .93 .94 .95 .96 .97T/Tc

.98I I

.99 I.OO

FIG. 10. Comparison of theoretical results and ex-perimental data on the temperature dependence of thelong-range order for PAA and PAP.

Calculations of the entropy discontinuity at thephase change were made in Sec. IIIA based on thesimplest potential function, Eq. (16). The resultsfor z=4, 6, 8, 10 are presented in Table I. 4S/Nkranges from about 0.2 to 0.35. Experimentalvalues of 4S/Nk can be computed from compila-tions of latent heat and critical temperature data. "Representative values of AS/Nk include 4-methoxy-benzylidene-4'-n butylaniline (MBBA), 0.16; 4-ethoxybenzylidene-4'-n butylaniline (EBBA), 0.29;4-butoxybenzylidene-4'- ethylaniline, 0.15; 4-cyanobenzylidene-4'-n- octyloxyaniline (C BOOA),0.27; anisylidene-p- aminophenylacetate (APAPA),0.39; anisylidene-p- aminophenyl-3- methylvalerate,0.15-0.3; 4-n-pentyl-4'-cyanobiphenyl (PCB),0.155-0.3; 4-n-nonly-4'- cyanobiphenyl, 0.15-0.5;4-n-pentoxy-4'-cyanobiphenyl, 0.15; 4-4' di-methoxyazoxybenzene (PAA), 0.12; 4-4'-dieth-oxyazoxybenzene (PAP), 0.34; and 4'-4'-di-n-heptyloxyazoxybenzene, 0.25-0.4. It is seen thatthe measured values are in good accord with thosegiven by the calculations.

Calculations of the temperature dependence of thelong-range-order parameter (P,) were consideredin Sec. III A and III B. We have made a comparisonof our theoretical results with experimental data"on PAA and PAP using the potential function ofEq. (23). A graphical compa. rison is shown in Fig.10. We have arbitrarily chosen z=6. The constantvp is then effective ly e lim inated by exp re ssing boththeory and experiment in reduced temperatureunits, T/T„and forcing the calcula, ted and ex-perimental critical temperatures to agree. Thisleaves n4 as the only remaining parameter to beadjusted. Excellent fits to the data, were obtainedby choosing @4=0 in the case of PAA, and n4 ——3

1892 PING SHENG AND PETER J. WOJTOWICZ 14

C z dT, dT4Nk 2 d(kT/)' d(,!Tl',) )

' (36)

Figures 11and 12 display the comparisons oftheoretical and experimental" orientational heatcapacities for PAA and PAP, respectively. Theonly fitting done for these figures was the estab-lishment of the base line for the experimentalpoints. That is, a roughly constant contributionto the measured heat capacity (representing thenonorientational components) was subtracted fromeach of the data points. The magnitude of thissubtracted portion was chosen so that the lowesttemperature data point in each case agrees withthe theoretical value. The agreement betweentheory and experiment is reasonable consideringthat no further adjustment of potential parameterswas made. In particular, the theory is seen toaccurately predict the width of the transition re-gion. A more quantitative agreement on the lowtemperature side of T, could certainly be obtained

18

I6-

l4-

12-

~II

IO-CR 8-

4-

in the case of PAP. A more precise fitting couldhave been made by the simultaneous adjustmentof z and a4 in a rms procedure and by taking intoaccount the temperature variation of vo. We donot feel, however, that such embellishments areinstructive considering the approximations al-ready made in arriving at the potential function,Eq. (23).

The most stringent test of any theory of orderedphases is the comparison of calculated and mea-sured heat capacity. Having determined the valuesof the potential parameters for PAA and PAP fromthe long-range-order data we can now compute theheat capacities without any further adjustments inthe constants. For the potential function, Eq. (23),the heat capacity is obtained from Eq. (14) bydifferentiation

24—

2I—

l8—

15

Rl2—

PAP (2 =6, a4 = I/3)

~ ~

0 I 1 I ) I I

.92 .93 .94 .95 .96 .97 .98 .99T/Tc

I I I

1.00 I.OI I.02 I.O3 I.04

FIG. 12. Comparison of theoretical results and ex-perimental data on the temperature dependence of theorientational heat capacity of PAP,

by simultaneously fitting (P,) and C data throughthe adjustment of z and n4 and taking into accountthe temperature variation of v, . The data on thehigh-temperature side of T„however, cannot beaccounted for quantitatively. As is well known

from the example of magnetism, "the constantcoupling theory cannot be accurate above thetransition temperature where long- range orderhas disappeared. In this regime the mean- fieldcomponent of the potential vanishes and the po-tential function reduces to that of isolated inter-acting pairs. A theory of isolated interactingpairs is, of course, incapable of modeling a highlyinteracting cooperative system.

The theory presented in this paper has beenbased on a model in which the interaction poten-tials a.re assumed to be volume (temperature) in-dependent (constant vo). An immediate consequenceof this restriction is the absence in the theory ofthe small experimentally observed volume dis-continuity at the phase transition. The absenceof the volume discontinuity then influences themagnitude of the theoretical latent heats and en-tropies of transition. In addition, the neglect ofthe volume (temperature) dependence of the inter-actions influences the precise temperature de-pendence of the order parameters and thermo-dynamic functions. Thus, though our calculationsare in reasonable agreement with experiment, athorough and more realistic fitting of experimentaldata will require the inclusion of the volume (tem-perature) dependence of the interactions potentialsin the formal development of the theory.

I

.9I .92 .93 .94 .95 .96 .97 .98 .99 I.OO l.02 I.03T/ Tc

APPENDIX

FIG. 11. Comparison of theoretical results and ex-perimental data on the temperature dependence of theorientational heat capacity of PAA.

Here we show that at any particular temperaturethe value of PF given by Eq. (15), with the o~ de-termined from Eq. (12), is unique up to a constant.

14 CONSTANT-COUPLING THEORY OF NEMATIC LIQUID CRYSTALS 1893

bottom of the trough is traced out by the loci onthe O,-P plane as shown in Fig. 13. Suppose nowwe want the value of PF at the point P. Let thevalue of PF at P=O, a, =0 be a constant C. Thevalue of PF at P is then given by

PF(atP) =C+ o, + P dS, (A1)' s(PF) - s(PF) - . -

o

where the integral is along the path indicated byarrows in Fig. 13. a„Pdenote unit vectors along the0, and P axes, respectively, and dS is a line ele-ment along the path of integration: P df= dP,0, .dS = do~. Since on the path of integration,

s(pF) 0 s(pF)80, ' BP

(A2)

as discussed in Sec. IIC, it immediately followsthat

FIG. 13. Schematic representation of the loci of thesolutions to the self-consistency condition, Eq. (12).The arrows indicate the path of integration in Eq. (Al).

PF(at P) = C+ E[oz(P), P] dP.0

(A3)

Figure 13 shows a schematic drawing of the lociof the solutions to t;he self-consistency relation,Eq. (12). For simplicity of illustration we haveonly shown the loci in the a,-P plane. However,the arguments presented in the following areequally applicable to the general case. We notein Fig. 13 that o, = 0 is a solution at all temper-atures and that two more branches of solutions ap-pear only below a certain temperature. The free-energy function, PF, can now be defined on the a,-Pplane as a surface whose distance above or belowthe a,—P plane at any particular point gives thevalue of pF at that point. On this free-energysurface there is a trough formed by the localminima of the free energy. The position of the

Here P~ is the value of P at point P as shown inFig. 13, and o~(P) denotes the values of the o~determined by Eq. (12) [o2(P) gives the loci on the

o,-P plane as shown in Fig. 13]. Since the valueof E is well defined at every point of the loci, Eq.(A3) shows that up to a constant, the two con-ditions as given by Eq. (A2) uniquely determinethe values of PF on the loci. Since the function PFas given by Eq. (15) satisfies these two conditions,this proves our assertion.

Note that we have only proven the uniqueness ofthe PF values on the loci of the solutions to Eq.(12). Indeed, different functions of PF may befound which have identical values on the loci [andtherefore satisfy the two conditions given byEq. (A2)] but differ elsewhere on the o,-P plane.

'L. Onsager, Ann. N. Y. Acad. Sci. 51, 627 (1949).~P. Sheng, RCA Rev. 35, 132 {1974).3W. Maier and A. Saupe, Z. Narturforsch. A 14, 882

(1959); 15, 287 (1960).P. J. Wojtowicz, RCA Rev. 35, 105 {1974).

~S. Chandrasekhar and N. V. Madhusudana, Acta Cryst-allogr. A 27, 303 (1971).

R. L. Humphries, P. G. James, and G. R. Luckhurst,J. Chem. Soc. Faraday Trans. II 68, 1031 (1972).P. J. Wojtowicz, RCA Rev. 35, 118 (1974).G. Lasher, Phys. Rev. A 5, 1350 (1972) ~

9P. A. Lebwohl and G. Lasher, Phys. Rev. A 6, 426(1972); 7, 2222 (1973).P. W. Kasteleijn and J. Van Kronendonk, Physica(Utr. ) 22, 317 (1956).B. Strieb, H. B. Callen, and G. Horwitz, Phys. Rev.130, 1798 (1963).

R. J. Elliot, J. Phys. Chem. Solids 16, 165 (1960).' H. B. Callen and E. Callen, Phys. Rev. 136, A1675

(1964).'4H. B. Callen and E. Callen, J. Phys. Soc. Jpn. 20,

1980 (1965).' E. Callen and H. B. Callen, J. Appl. Phys. 36, 1140

(1965).' J. S ~ Smart, Effective Field Theories of Magnetism

(Saunders, Philadelphia, 1966), Chap. 5.~YJ. C. Raich, R. D. Etters, and L. Flax, Chem. Phys.

Lett. 6, 491 (1970).R. G. Priest, Phys. Rev. Lett. 26, 423 (1971).

BJ. A. Pople, Proc. R. Soc. A 221, 498 (1954).M. E. Rose, Elementary Theory of Angular Momentum(Wiley, New York, 1957).

'T. Oguchi, Prog. Theor. Phys. 13, 148 (1955).N. V. Madhusudana and S. Chandrasekhar, Solid State

PING SHENG AND PETER J. %OJTO%ICZ

Commun. 13, 377 (1973).3T. D. Schultz, Mol. Crystallogr. Liq. Crystallogr. 14,147 (1971).

24K. B. Priestley, P. J. Wojtowicz, and P. Sheng, Intro-duction to Liquid Crystals (Plenum, New York, 1975),p. 351.

25Data of S. H. alarum and J. H. Marshall, J. Chem.Phys. 44, 2884 (1.966); A. Saupe, Angew. Chem. (Inter.Ed.) 7, 97 (1968); S. Chandrasekhar and N. V. Mad-husudana, J. Phys. C 30, 24 (1969); as quoted in Ref. 5.

2~Data of H. Arnold, Z. Phys. Chem. 226, 146 (1964), asquoted in Ref. 5.